You may have a Vacuum leak in one of the switches that enable the idle up with a/c or lights on !
I know you have fuel injection, but believe you still have some of the idle up components shared with all corollas that cause idle up with a/c and/or lights turned on.. it has an '
idle up' circuit.
I am very interested in the lights switched on and idle up myself. I while back immediately after I rebuilt my carburetor the idle up with the lights on stopped working. for the life of my I cannot figure out what I messed up. The A/C idle up still work great.
There are idle up vacuum lines and switches on the firewall with the vacuum lines running to them. It is sort of a boolean type operation where a combination of vacuum lines opened by the lights enable a small bit of air to enter through these switches allowing the idle to increase a small amount. The a/c has a vacuum line that goes to the firewall ont he passenger side and a metal vacuum switch on the back, under the carburetor that pulled a diaphram that in turn pull some linkage which turns the idle up. This is on the passenger side of the carb.
Since your problem lies with both a/c and lights not enabling the the idle up, it is at a common point with both.
I am intrerested in seeing your solution.
BTW, since mine in a 1989 Corolla, and yours is much newer, I can only assume you have the same idle-up stuff, but you have additional parts mine doe snot have since yours has Fuel Injection instead of a carb. Yours has a idle control valve my carb model does not. It is called Idle Air Control Valve.